Space Curves

In 3-space we take advantage of the usual vector algebra operations available on $\mathbb{R} ^3$ to study the curvature (departure from linearity) and torsion (departure from planarity) of curves in space. Since we are interested in curves with nonzero speed everywhere, we can always reparametrize to have unit speed; then the parameter coincides with arc length along the curve, often denoted by s.

Curvature, torsion and the Frenet-Serret equations The curvature of the unit speed space curve

$$\beta:[a,b]\rightarrow \mathbb{E} ^3$$ (15)

is the norm of its acceleration

$$\kappa[\beta](s)=\Vert\beta''(s)\Vert$$ (16)

It is easy to show that the velocity vector $\beta'$ is perpendicular to the acceleration vector $\beta''$ by differentiating $(\beta'\cdot\beta')=1.$ So if we take their cross product we get a vector perpendicular to both; we have only three dimensions and so the derivative of the new vector must be expressible in terms of the others. In this way, three, mutually perpendicular unit vectors $\{T,N,B\}$arise at each point: $T=\beta', \, N=T'/\kappa \ {\rm and} \ B=T\times N.$ These vector functions along the curve $\beta$ with curvature $\kappa$ are controlled by the famous
Frenet-Serret equations for unit-speed curves:

$$\kappa N \ \ \ \ \ {\rm recall \ that \ we \ have} \ \kappa >0$$ T' = (17)

$$-\kappa T + \tau B$$ N' = (18)

$$-\tau N$$ B' = (19)

Here, N is the principal normal, B is the binormal and $\tau$ is the torsion. $\{T,N,B\}$ is called the Frenet frame field along $\beta,$ and consists of three mutually perpendicular unit vectors--a triad that moves along the curve with T pointing always forward.

For a regular curve $\alpha$ with arbitrary speed $\sqrt{\alpha'\cdot\alpha'}=\vert\vert\alpha'\vert\vert=v>0,$ we have the
Frenet-Serret equations for arbitrary-speed curves:

$$v\kappa N \ \ \ \ \ {\rm recall \ that \ we \ have} \ \kappa >0$$ T' = (20)

$$-v\kappa T + v\tau B$$ N' = (21)

$$-v\tau N.$$ B' = (22)

  

Figure 1: A torus knot: This tube is a thickened embedding of a circle that has been mapped onto the surface of a torus; note the twisting of the ruling lines, showing high torsion when there is rapid departure from a plane.

Exercises on Frenet-Serret equations
Here, $\beta$ is the unit speed curve in equation (15).

1.

Show that the helix

$$\gamma:[0,10]\rightarrow \mathbb{E} ^3: s\mapsto (2\cos(\frac{s}{\sqrt 5}),2\sin(\frac{s}{\sqrt 5}),\frac{s}{\sqrt 5})$$ (23)

is a unit speed curve and has constant curvature and torsion.

2.

Why do we always have $\kappa[\beta]\geq 0?$

3.

For all s, $\beta''(s)\cdot\beta'(s)=0;$ so the acceleration is always perpendicular to the acceleration along unit-speed curves. What about $\alpha'(t)\cdot\alpha''(t)$ on arbitrary speed curves?

4.

Derive the Frenet-Serret equations for an arbitrary-speed regular curve $\alpha$ and show that the following hold for a curve $\alpha$ with speed $\sqrt{\alpha'\cdot\alpha'}=\vert\vert\alpha'\vert\vert=v>0$:

$$T=\alpha'/v, \ \ \ \ \ N B\times T, \ \ \ \ \ B=\frac{\alpha'\times\alpha''}{\vert\vert\alpha'\times\alpha''\vert\vert}$$ = (24)

$$\kappa \frac{\vert\vert\alpha'\times\alpha''\vert\vert}{v^3}$$ = (25)

$$\tau \frac{\alpha'\times\alpha''\cdot\alpha'''}{\vert\vert\alpha'\times\alpha''\vert\vert^2}.$$ = (26)

5.

Viviani's curve² is the intersection of the cylinder (x-a)²+y²=a² and the sphere x²+y²+z²=4a² and has parametric equation:

$$\alpha:[0,4\pi]\rightarrow \mathbb{E} ^3:t\mapsto a(1 + \cos t,\sin t, 2\,\sin \frac{t}{2}).$$

Show that it has curvature and torsion given by

$$\kappa(t)=\frac{{\sqrt{13 + 3\,\cos t}}} {{a\left( 3 + \cos t... ...d} \ \ \ \tau(t)=\frac{6\,\cos \frac{t}{2}}{a(13 + 3\,\cos t)}.$$

6.

Investigate the following curves for n=0,1,2,3

$$\gamma:[0,2\pi\sqrt{6}]\rightarrow \mathbb{E} ^3: s\mapsto (\... ...frac{s}{\sqrt 6}),\frac{\sqrt{3}}{2}\sin(\frac{n s}{\sqrt 6}))$$ (27)

7.

Show that for all $\theta\in [0,2\pi]$ the matrix

$$R_z(\theta)=\left (\begin{array}{ccc} \cos\theta &\sin\theta &0 \\ -\sin\theta&\cos\theta &0\\ 0&0&1\end{array} \right)$$

when applied to the coordinates of a curve in $\mathbb{E} ^3$rotates the curve through angle $\theta$ in the (x,y)-plane, that is, round the z-axis. Find a matrix $R_y(\theta)$representing rotation round the y-axis and hence obtain explicitly the result of rotating the curves in the previous question by 60^o round the y-axis.

8.

On plane curves, $\tau=0$ everywhere and we sometimes use the signed curvature $\kappa2,$ defined by

$$\kappa2[\alpha](t)=\frac{\alpha''(t)\cdot J\alpha'(t)}{\Vert\... ...:\mathbb{R} ^2\rightarrow \mathbb{R} ^2 : (p,q)\mapsto (-q,p).$$ (28)

We call $1/\kappa 2[\alpha]$ the radius of curvature of $\alpha.$ Find the radius of curvature of some plane curves.

9.

(i) Find two matrices, R_y and R_z from SO(3)which represent, respectively, rotation by $\pi/3$ about the y-axis and rotation by $\pi/4$ about the z-axis; each rotation must be in a right-hand-screw sense in the positive direction of its axis. Find the product matrix R_yR_z and show that its transpose is its inverse.
(ii) By considering (R_yR_z)^-1, or otherwise, show that the curve

$$\gamma:[0,\infty) \rightarrow \mathbb{E} ^3: t\mapsto (\frac{... ...ac{\sqrt{3}}{\sqrt{2}}\cosh{t/2} - \frac{\sqrt{3}t}{2\sqrt{2}})$$

lies in a plane and find its curvature function and arc length function.

10.

Vertical projection from $\mathbb{E} ^3$ onto its xy-plane is given by the map

$$\pi:\mathbb{E} ^3\rightarrow\mathbb{E} ^3:(x,y,z)\mapsto(x,y,0).$$

A unit speed curve $\beta :[0,L]\rightarrow \mathbb{E} ^3$ lies above the xy-plane and has vertical projection

$$\pi\circ\beta:[0,L]\rightarrow\mathbb{E} ^3:s\mapsto (\frac{s}{2}\cos(\log{s/2}),\frac{s}{2}\sin(\log{s/2}),0).$$

Find explicitly a suitable $\beta$ and for it compute the Frenet-Serret frame, curvature and torsion.

11.

Investigate a selection of named curves from Gray [6,1].

Classification of curves Regular space curves with nonzero curvature are classified by their curvature and torsion, up to a Euclidean transformation (translation plus reflection and/or rotation):

Theorem 6.1 (Fundamental Theorem of Space Curves)   Two space curves defined on the same interval with the same torsion and nonzero curvature can be transformed into one another by application of a translation and a Euclidean transformation.

There is a reduced version for plane curves:

Theorem 6.2 (Fundamental Theorem of Plane Curves)   Two regular plane curves defined on the same interval with the same curvature $\kappa2,$ can be transformed into one another by application of a translation and an orthogonal transformation.

Gray [6] proves the classification theorems and studies applications in detail, giving many examples and a Mathematica algorithm for drawing a curve in $\mathbb{E} ^3$ with specified curvature and torsion.